Question
measure of two sides of a triangle is 5.2 and 8.1 what is the range for the third angle
Answers
GPT-4o mini
To find the range for the third side of a triangle when two sides are given (5.2 and 8.1), we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, and the length of any side must be positive.
Let \( a = 5.2 \) and \( b = 8.1 \). Let \( c \) be the length of the third side. The triangle inequality gives us the following inequalities:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting the known values into these inequalities:
1. \( 5.2 + 8.1 > c \)
\( 13.3 > c \) or \( c < 13.3 \)
2. \( 5.2 + c > 8.1 \)
\( c > 8.1 - 5.2 \)
\( c > 2.9 \)
3. \( 8.1 + c > 5.2 \)
\( c > 5.2 - 8.1 \)
\( c > -2.9 \) (This condition is always satisfied since side lengths must be positive.)
Combining the results from the first two inequalities, we have:
\[ 2.9 < c < 13.3 \]
Thus, the range for the third side \( c \) of the triangle is:
\[ (2.9, 13.3) \]
Let \( a = 5.2 \) and \( b = 8.1 \). Let \( c \) be the length of the third side. The triangle inequality gives us the following inequalities:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting the known values into these inequalities:
1. \( 5.2 + 8.1 > c \)
\( 13.3 > c \) or \( c < 13.3 \)
2. \( 5.2 + c > 8.1 \)
\( c > 8.1 - 5.2 \)
\( c > 2.9 \)
3. \( 8.1 + c > 5.2 \)
\( c > 5.2 - 8.1 \)
\( c > -2.9 \) (This condition is always satisfied since side lengths must be positive.)
Combining the results from the first two inequalities, we have:
\[ 2.9 < c < 13.3 \]
Thus, the range for the third side \( c \) of the triangle is:
\[ (2.9, 13.3) \]